Mathew Reynolds^a, Michael Reynolds^b, Samer Adeeb^a, Tarek El-Bilay^c,

Abstract:

Bone growth is a complex process that is controlled by a multitude of mechanisms that are not fully understood. Of these mechanisms the growth of bone has been argued to be stimulated by mechanical stimulus as well as a separate time component of growth. Current methods to measure the growth of bone have focused on cadaver studies of different ages, or two dimensional radiographs. The purpose of this study was to explore a technique for quantifying the three dimensional growth of an adolescant human mandible over the period of three years (2007 to 2010). The CBCT (cone beam computed tomography) taken for this patient was part of the orthodontic records. The data was then used to build 3-D models which allowed the comparison of the resulting surface meshes. The results of this work is in agreement with previously reported data from human cadavers. The presented method can provide a new tool to evalaute and potentially predict the patients’ mandibular growth.

*At publication time this will be the extent of the article viewable at this location.

Introduction:

Bone growth is a complex process involving biochemical and physical stimuli that are not yet fully understood. The process is further complicated considering that bone is a dynamic structure undergoing significant remodelling over the course of its life. Remodelling is the process of the deposition and resorption of the bone through specialized bone cells. Bone growth results from a slightly faster rate of deposition over resorption, resulting in a gradual increase in size as time progresses  (Martini, et al., 2008). Bone growth occurs by endochondoral (at areas of growing cartilage) or by intramembranous like through the periosteum (outside covering of the bone) or the endosteum (inside covering of the bone). The endochondoral bone growth at centers is responsible for lengthening bones (like the epiphyseal plates). The driving mechanisms of bone growth likely involve everything from diet to physical stress including a time dependent component of growth (Adeeb, et al., 2009). It has also been experimentally shown that varied mastication stresses due to the consistency of foods “markedly affect the manidublar condylar cartilage growth and mandibular morphology” (Enomoto, et al., 2010). The evaluation of the growth of the mandible is very complex due to its geometric shape, and the significant bone drifting (Enlow, et al., 1982)

The drift of bones is caused from primary and secondary displacements. Primary displacements describe relative movement of a bone due to its normal growth. On the other hand, secondary displacement is due to the movement of a bone caused by enlargement of neighbouring bones and/or their soft tissues. The mandible is subjected to significant primary displacements due to its growth displacing the relative position of the symphysis. Secondary displacement is also significant due to the growth of the maxillary and temporal bones. The drifting of the mandible by secondary displacement was not taken into account during this study as the evaluation and alignment method of the two mandibles represent changes in overall shape and not changes in its shape with respect to anatomic position.

A  medical software packages (MATERIALIZE: MIMICS version 13, Leuven Belgium) allow the creation of models in the form of STL (sterolithography) from threshold segmented images. The STL format is a popular format due to ease of readability and physical meaning of the code as shown by (Wang, et al., 2010). Corresponding STL file  are then compared using software techniques utilized for rapid prototyping model verification.  This technique allows the production of a 3D deviation map between the two models. The advantage of using this technique is that the method represented is capable of representing the direction and magnitude of bone growth. This numerical growth data is the first known in-vivo numerical growth data done in 3D. The advantage of using 3D models to represent medical images is paramount, as discussed by  (Sabourin, et al., 2010) and  (Swann, 1996).

Materials and Methods

Computerized Tomography Scans

The study consisted of three consectuive CT scans of an individual paitent that originally was taken as a part of the orthodontic records. The three scans were taken  once every year. Three dimensional models were created from CT images using the software package: Materialise MIMICS, which was used to export a STL  file.

Preliminary 3D Geometries

The CT images used are 12 bit (4096 grey values possible per pixel) pixel maps that illustrate the X-ray attenuation coefficient of the tissue (Bankman, 2009). Each mandible CT data set used in this study consisted of 440 slices 0.3mm apart, per data set, with an in plane pixel size of 0.3mm – yielding a voxel (3D version of 2D pixels by stretching the pixels over their associated slice distance) volume of 0.027mm³. The program MIMICS initially scales the images to the Hounsfield scale, which sets the attenuation of water to zero. Using the Hounsfield scale the minimum pixel value is -1024, representing the attenuation of air. The estimated pixel values using Hounsfield units (HU) that represents trabecular and cortical bone are (150-250 HU) and (251-2100 HU) respectively. The two latest CT data sets obtained from identical machines were imported into MIMICS and appropriately scaled to the Hounsfield scale. To ensure scientific conformity the same pixel threshold values of 226 HU to 2042 HU is used when threshold segmenting the images similar to the method illustrated by Stratemann, et al., (2010). Threshold segmentation creates a highlighted region of interest called a mask (Figure 1). The initial estimated mask volume of the 2008 scan is 34.7 cm³ while the estimated mask volume of the 2009 mask is 34.8 cm³. The average grey values of the 2008 and 2009 masks are both 838 HU, while the respective standard deviations ofthe masks are 459 HU and 435 HU. The two CT scans were affected from artifacts surrounding a lower lingual arch (a dental appliance attached to the mandibular teeth). Artifacts of this nature are created by the photoelectric effect due to the high relative atomic number of the appliance. This affects the apparent attenuation of the surrounding tissue because most of the beam is removed by photoelectric effects. Therefore there is missing information when the CT data is back projected causing gross anomalies. Thus in order to create a 3D geometry that is a true representation of the mandible, the teeth and surrounding region affected by the anomalies are removed from the models.

MATERIALIZE: MIMICS was then used to calculate a 3D object from the two associated masks using high quality settings that preserve the integrity of the models (see figure two). A grey value interpolation scheme was utilized because of its accounting for partial volume effects of the voxels. This produces a generally more accurate model vs. a contour interpolation method in which contours are drawn on the mask boundaries on each slice. The contours would then be filled in-between using linear interpolation, which ignores any partial volume voxels between the contours. 3D models are then smoothed (with settings that preserve the shape and size of local geometries) in order to reduce noise in the model. Once acceptable 3D models of the two specimens are created the models are exported to STL format.

3D geometry analysis

The two STL files were imported into GEOMAGICS STUDIO/QUALIFY Version 11, North Carolina, a 3D suite built around in order to optimizing and comparing 3D models to a reference model. Within the STUDIO suite the STL file is optimized by removing anomalies in the form of protruding vertices and localized holes. Typically the anomalies are initially created due to a combination of noise in the CT image and the interpolation algorithm used to create the triangulated mesh. At this step any excessive deformations in the area of the teeth affected by the X-ray artifacts were removed manually. Following this the mesh was rewrapped with new triangles using a built in algorithm, this ensured the triangles on both models were equal in size – important for comparing the models.

After optimization the models were imported into GEOMAGICS QUALIFY. To compare the two mandible models the 2008 model was set as the reference model, and the 2009 model is used as a comparison. The two models are aligned using a built in algorithm for best fit. The algorithm sets the 2008 model as a fixed model, and works in two stages by floating the 2009 model (Figure 3). First stage fitting uses a small number of points (300) from similar locations on each model to make a gross movement of the models. Second stage uses a larger sample of points (1500) to make fine adjustments to best fit the model. The reported average error from the alignment is 0.452mm. Using a sensitivity analysis it was determined that increasing the sample size of the best fit algorithm does not significantly improve the fit of the two models (0.01mm improvement in error for every extra 1000 points). Another method to align the models is to use a list of landmark points as according to (Stratemann, et al., 2010) and (Liu, et al., 2010), or to only quantify a series of measurements between models (Enomoto, et al., 2010). It was decided in this study to use a best fit alignment because it is the entire surface is being compared, and not the relative movement of landmarks. The artifacts affecting the region of the teeth have a small effect on the alignment of the two models. This effect is thought to be relatively small due to the relatively small surface area of the models affected by the anomalies. Following the alignment the two models are compared by an illustration of the 3D deviation of the two models. A 3D deviation (Figure 4) is computed by measuring the orthogonal distance from the vertices of the reference object (2008 model) to the float object (2009 model). To ensure realistic computational time the max result point count was limited to 2 million data points. Table 1 illustrates the output parameters from the 3D comparison. It should be noted that the max deviation of 4.34mm is located at the teeth region where the artifacts are present. The large standard deviation is again thought to arise from the artifacts present around the teeth. The 3D comparison colour model was the exported as a VRML file (Virtual Reality Modeling Language), this format allows the embedment of colour within the 3D mesh.

Table 1: 3D comparison parameters output from Geomagics: Qualify

Comparison

Liu (2010)  reported several two dimensional parameters used to describe the growth of the mandible by defining landmarks using planes and bisected angles. These landmarks have been used in order to compare to literature, as well as to showcase some of the data possible with this kind of analysis. Table 2 presents landmarks and succinct instructions on how to find them as reported by Liu (2010).

Table 2: LANDMARKS USED TO EVALUATE THE GROWTH OF THE MANDIBLE (LIU, ET AL., 2010)

3D Printing

The growth study was followed up with the creation of a 3D physical model, a novel and unique way to illustrate the growth of the mandible. A growth map from the aforementioned study was printed on a colour 3D printer, allowing a “graspable” physical model representing the growth of the mandible over a year. This was accomplished using the VRML file, which allows for a 3D  printer toprint the surface mesh with associated colours. The production of a physical model has great benefits for the use of explaining and conveying information.

The 3D printing used is a Spectrum Z™510 colour 3D printing system manufactured by Z corporation (www.zcorp.com). The printer works in the same way most 3D printers operate, in which a printer head selectively hardens thin layers of plaster duster with a sugar-water binding agent. The innovation of colour 3D printing comes in the binding agent, in which four separate print heads contain four binding agents: clear, cyan, magenta and yellow. Selective use of binding agents to harden regions of plaster allows the printing of 24 bit full colour models. Each layer of the model hardened by the binding agents is 0.1mm thick, with a max resolution of 600 x 540 dpi (www.zcrop.com).

Results

The results of our study include a 3-D dimensional virtual model of the 3-D deviation of the jaw between 2008 and 2009 (Figure 5[t5] ). In addition to the virtual model a 3D physical model (Figure 6) illustrating the same growth map allows for easy viewing and in depth analysis.

Figure 7 shows a comparison of our growth map with predicted growth from literature (Enlow, et al., 1982). Recorded growth is shown by a series of arrows showing the relative shape movement of the mandible as it grows, the size of the arrows representing the magnitude of growth over time.

Figure 8 shows a comparison of the 2008-2009 cross sections compared with literature cross sectional growth data. From the figure we can see general conformity with the models and of the literature values (Enlow, et al., 1982) and (Dixon, et al., 1997). Figure 9 illustrates the comparison of the change of the ramus, condyle and coronoid process. As illustrated our data matches predicted growth value, with relevant numerical magnitude. Maximum recorded growth (ignoring the areas affected by the artifacts) between the 2008 and 2009 models is 1.67 mm on the condyle while the maximum resorption is -1.0 mm adjacent to the symphysis (most anterior part of the mandible) .

Cutting of the Ramus region including the condyle and coronoid process regions (Figure 10) region shows an overall decrease in volume of -0.5%. The overall width of the mandible, measured from the inside of the condyle, increased by 0.49mm – an increase of 2.8 %. A change in this size is relatively small, likely due the fact our model is representing a stage where the mandible is approaching maturity and thus experiencing a deceleration of growth (Liu, et al., 2010).

Table 3 is data obtained by comparing the distances discussed by Liu (2009). In addition to the data presented in table two there is another angle formed between the corpus and the symphysis plane that remained constant at approximately 85 degrees.  Figure eleven illustrates the important parameters superimposed on an elevation view of the 2009 mandible model, showing the landmarks used to extract the data.

Table 3: Results of various measurements on the mandible comapring models from 2009 and 2010

Discussion

The first scan (2007) and second scan (2008) were completed on different CT machines with different contrasts and thus different algorithims were required to formulate a model for each scan. Using different algorithims likely impacted the accuracy of the models, and thus the 2007 scan was used only as a pilot study. The third scan (2009) was done using the same CT machine as the 2008 model, and thus the same modeling algorithim was used to create a three dimensional model.

The regions affected by the artifacts were removed by selecting the voxels in the area of the artifact. Therefore growth data in this region is no longer valid due to the arbitrary cutting, and volume data utilizing the entire mandible has no scientific relevance. It is expected that due to the rapid relative growth and movement of teeth that data in this region would be invalid regardless. This is the reason why the virtual and physical model illustrates unrealistically large growth predictions on the upper portion of the mandible body.

Our model of mandibular growth that was created from the CT scans is congruent with literature on mandibular growth (Enlow, et al., 1982) and (Dixon, et al., 1997) . The model has the same regions of growth as predicted from literature, with an advantage that our model carries numerical meaning with its colour map. Furthermore our model represents the growth of a single patient – showcasing an important distinction between the recorded growth between multiple cadaver specimens.

The growth discussed in our study is the 3D deviation from two CT scans aligned using a best fit  algorithm and thus  our growth map does not take into account the drift affects from primary and secondary displacements. Therefore our model represents the overall shape change of the mandible, and is not a representation of how the mandible grows in its anatomical position. This is evident in examining the condyle region of our printed mandible, in which you can see relative significant growth. An outward projection of the mandible is impossible due to the location of the temporomandibular joint (TMJ) in the glenoid fossa. The actual growth would be that the condyles stayed in the same relative positions and the ramus extended lowering the level of the body of the mandible. When comparing to literature this is an important distinction, as the common source of relative growth information of the mandible is based on cadaver studies averaged over multiple patients illustrating the shape change of the mandible – thus making it easy to compare with our study.

The maximum deposition being greater than the maximum  resorption rate is an indication of growth. Maximum resorption and deposition are relatively small compared to Liu (2009), but this is likely due to the decelerated growth in our studied jaw after the first growth spurt (age 9-10). The small decrease in ramus volume is likely due imperfect modeling and scanning artifacts – as it is expected that the volume would remain the same. A constant volume is expected due to the ramus remodeling outwards and adding material to the symphysis allowing the condyles to grow laterally . Our study showed that the condyles grew laterally (2.8%) over the course of the year – this could be due to the fact that the mandible is growing to match the growth of the maxilla (Dixon, et al., 1997).

Dixon, et al. (1997) argues the reason that the depth of the mandibular notch increases is due to the gradual increase of forces on each side of the notch. The coronoid process is pulled upwards due to the temporalis muscle causing a general elongation of the process. While the condyle is loaded through the TMJ joint, and is thought to elongate as response to the stress. The general understanding that the forces of mastication have a large impact on the shape of the mandible are thoroughly discussed by (Dixon, et al., 1997), (Enlow, et al., 1982) and (Enomoto, et al., 2010).

The gonial angle (mandible angle) remained constant at 128 degrees throughout the course of the study (1 year). The expected result is that the gonial angle will decrease as time progresses as shown by (Dixon, et al., 1997) and (Enlow, et al., 1982). The expected averaged rate is less than a degree a year (Liu, et al., 2010) and it is therefore likely that our study does not offer the resolution required to track an angle of that magnitude. A gonial angle of 128 degrees reflects an average angle, as a typical gonial angle varies from 140 degrees at birth to 120 at maturity (Dixon, et al., 1997).

The Corpus Length increased by 0.3mm a change of 0.5% while the ramus height increased by 0.5mm, a change of 1.1% over the course of the study. No known values at this age have been found for these measurements although the increasing trend has been reported in literature (Liu, et al., 2010). The overall Length defined earlier increased by a 0.6mm. This is expected as it is the hypotenuse of the corpus length and ramus height, both of which increased in dimension.

The importance of using graspable objects instead of Virtual Reality objects on a computer screen are summarized by  (Rengler, et al., 2010),  (Wang, et al., 2010) and  (de Beer, et al., 2007). The embedment of colour on our 3D model illustrating the growth between the two models is the first time an accurate physical representation of mandible growth has been accomplished. Previous growth maps of the mandible exist only as 2D pictures showing arrows indicating the relative growth. Our 3D model has the advantage of having an infinite number of orientations, with a colour map illustrating the magnitude of growth in a region.

Conclusion

A model of the growth of a mandible is created by creating 3D files from two successive CBCT scans one year apart. The growth model created showcases the growth over the course of one year by comparing the 3D deviation of two models created from CT data. Recorded growth from literature (based on statistical averages of cadaveric specimens) illustrated the same trends as our growth model. The growth model thus not only showed accurate relative growth in a 3D form but provided numerical data.

The advantage of having accurate growth data with a convenient way to showcase the results has important clinical and academic relevance. Clinically making accurate growth maps from regular dental scans can show more in depth detail and early detection of problems. The creation of 3D models has been shown to be  a viable and powerful clinical technique; whether used for comparing 3D anatomic surfaces (Stratemann, et al., 2010), creating 3D representations for surgical problems (Sabourin, et al., 2010) and (Swann, 1996), streamlining the fitting and creation of medical implants (deBeer, et al., 2007). Furthermore Rengler, et al., (2010) discussed the value of creating physical models for use in the clinical setting. Clinically the aforementioned numerical details can be determined on a patient yearly and tracked, and compared with statistical normal’s leading to an in-depth analysis of a patient’s growth. Academically accurate growth measurements lead to insight in validating growth models, as well as defining normals for dental research.

References

Adeeb Samer and Herzog Walter Simulation of biological growth [Journal] // Computer Methods in Biomechanics and Biomedical Engineering. - 2009. - Vol. 12. - pp. 617-626.

Bankman Issac N Handbook of Medical Image Processiong and Analysis [Book]. - Amsterdam : Elsevier/ Academic Press, 2009.

de Beer Deon [et al.] Using RP to promote collaborative design of customised medical implants [Journal] // Journal of Rapid Prototyping. - 2007. - Vol. 13/2. - pp. 107-114.

Dixon Andrew D, Hoyte David A.N and Ronning Olli Fundamentals of Caniofacial Growth [Book]. - Flordia : CRC Press LLC, 1997.

Enlow Donald, Moyers Robert and Merow William Handbook of Facial Growth [Book]. - [s.l.] : W.B Saunders Company, 1982.

Enomoto Akikio [et al.] Effects of mastication on mandibular growth evaluatied by microcomputed tomography [Journal] // Journal of Orthodontics. - 2010. - Vol. 32. - pp. 66-70.

Liu Yi-Ping, Hehrents Rolf G and Buschang Peter H Mandibular Growth, Remodeling, and Maturaton Curing Infancy and Early Childhood [Journal] // Angle Orthodontists. - 2010. - Vol. 80. - pp. 97-105.

Martini Frederic and Nath Judi Fundamentals of Anatomy and Physiology Eighth Edition [Book]. - [s.l.] : Pearson Benjamin Cummings, 2008.

Rengler F [et al.] 3D printing based on imaging data: review of medical applications [Journal] // Journal of Rapid Prototyping. - May 2010.

Sabourin Marc [et al.] Three-dimensional steroradiographic modeling of rib cage before and after spinal rowing rod procedures in early-onset scoliosis [Journal] // Clinical Biomechanics. - 2010. - Vol. 25. - pp. 284-291.

Stratemann Scott A [et al.] Evaluating the mandible with cone-beam computed tomography [Journal] // American Association of Orthodontists. - April 2010. - pp. 58-70.

Swann S Integration of MRI and sterolithogarphy to build medical models: a case study [Journal] // Journal of Rapid Prototyping. - 1996. - Vol. 2. - pp. 41-46.

Wang Chung-Shing, Wang Wie-Hua A and Lin Man-Ching STL rapid prototyping bio-CAD model for CT medical image segmentation [Journal] // Computers in Industry. - 2010. - Vol. 61. - pp. 187-197.

http://www.ualberta.ca/~mpreynol/MimicsVideos/

University of Alberta, University of Calgary

Introduction

Ligaments are the supporting structure between two bones that serve to guide joint movements and maintain joint congruency (Frank et al. 1999.)  Ligament fibers tend to run parallel to the length but exhibit a corrugated appearance that differs in amplitude depending on location, thus offering non-homogenous strain behavior (Frank et al. 1999.)

Current strain measurement techniques in soft tissues are limited to strain gauges that offer limited range of valid data materials offering non-homogeneity (Phatak, 2007.) Digital Imaging Correlation (DIC) was used in this study on two different models to map strain for the specimans entire length.  DIC is a process that correlates points on deformed images with the same points on a reference image. This allows strain to be calculated in 3D on any surface.

First instance investigated was that of the Collateral Knee Ligaments in direct tension with both insertion sites intact. The second study was that of an MCL undergoing an in vitro simulated gait cycle.

Results

A tension test of an MCL is seen in figure 1. The results show higher strain based at the insertion site, shown by the red gradient of the major principle stress. The arrows on the diagram depict the direction of the major strain direction.

e1 of LCL

LCL e1

LCL e2

The results of the MCL during gait cycle shows non uniform strain distributions when under load. It is also observed that the MCL is under load only 3 times during gait.

LCL Direction of Displacement

Conclusions

The tension test of an MCL results in higher initial strain values at insertion sites as found by Kawada et al. (1999). Our study indicated principle strain directions tend to

run parallel, and at an angle to the surface of the ligament. Hirokawa  et al. discussed that  the principle strain lines depict the primary fiber direction. By observation of the ratio of minor by major principle strains it is observed that the volume is increasing at the interface, and decreasing at the mid substance.

The MCL under simulated gait cycle illustrates very different strains depending on degree of flexion in the joint. The strains range from high interface strains to higher mid substance strains depending on location in the gait cycle.

Area of Interest:

Structural, Fluid Mechanics and Biomedical Engineering

Objective:

To gain knowledge in the complex nature of Biomedical Engineering. The pursuit of this goal will involve the completion of higher education in the form of a Masters or Doctorate Degree and (or) Employment in an area of interest.

Education:

Bachelor of Science in Civil Engineering – Biomedical Specialization, (2010)

University of Alberta, Alberta Canada

Project Experience:

•     Structural Analysis of Biomedical Plates: Investigated repair techniques to a proximal humerus fractures using locking plate fixation techniques (publication pending).

•      Strain ligaments in Soft Tissues: Used a Digital Imaging Correlation to investigate surface strains in bovine Medial and Lateral collateral ligaments (Poster Presentation).

•     Growth Study of Human Mandible: Analysis of growth in an adolescent subject using Cad and FEA models generated from CT scans (Poster Presentation).

•     Accessibility and Universal Design: Database and resource management for outpaitents and interested parties. (www.homeforlife.info)

Background Courses Include:

Employment:

University of Alberta/Alberta Health Services, Edmonton, AB    May 20010-Aug 20010

• Lab cordinater for Biomechanics Lab.
• Supervise various research projects involving bovine soft tissue studies, cadevaric humeral fixation study and numerical growth study.
• Collection and publication of in depth examples of accessible design.

Associated Engineering, Edmonton, AB    May 2009-Aug 2009

• Surveyor involved underground utilities, lagoons, buildings, TOPO’s.
• Managed survey crew.
• Proficient in GPS (TOPCON), Total Station, and Optical Instruments.

EXH Engineering Services, Edmonton, AB    May 2007-Dec 2007

• Construction Management (Field and Office work).
• Took part in Bridge Inspections.

• Survey Rod Man for Road projects and TOPO’s.
• Experience with Nuclear Density Gauge and Materials Testing.

MDE Industrial Metal Deck, Edmonton, AB    July 2005-Aug 2006

• Labourer throughout high school.
• Promoted to drafting and design with project management duties.
• Worked on design of metal roofing systems.

Computer Skills:

– AutoCAD

- MS Office (Word, Excel, PowerPoint)

- Mimics, Geomagics

- Photoshop, Illustrator

- C++

- MATLAB, Maple,Mathematica

- Digital Imaging Correlation Software and Hardware systems

Volunteering:

- Scouts Canada Group Leader -06

- Grant Macewan Boot Camp Presenter -06/07

- U of A Civil Biomedical Open House Coordinator -08/09

- Apegga Lego Competition Judge -07/09

Professional Memberships:

- APEGGA (EIT) – Pending

- CSCE Student Member

Honours, Awards, Presentations:

- Jason Lang – 06/07/08/09

- Deans Research Award – 08/09/09

- NSERC Undergrad Summer Research Award – 08

- BME U/G Conference Attendance award – 08

- AIF U/G Biomedical Engineering Award- 09

- Leonard E Gads Memorial Scholarship – 09

- BME Conference Poster Presentation- 08/09

- CSCE Poster Competition (3^rd Place) – 08

- U of A: Engineering the Human Body Presentation -08

- NSERC Summer Presentation: Modelling the Human Body – 08

Publications:

- Biomechanical analysis of proximal humeral fixation using locking plate fixation with an intramedullary fibular allograft, Journal of Clinical Biomecahnics, May 2010

Craig Mathison,  Rey Chaudhary, Lauren Beaupre, Mathew Reynolds, Samer Adeeb, and Martin Bouliane

- Mandibular growth Investigation using 3-D volumetric evaluation (Pending), Journal of Computer Methods in Biomecahnics and Biomedical Engineering

Mathew Reynolds , Michael Reynolds , Samer Adeeb , Tarek El-Bilay

Dr. N Rajaratnam

Introduction

Blood is a specialized tissue that is made up of cells suspended in a fluid matrix. The typical human being has approximately 7% by mass of blood, which means the average male contains 5-6 litres of blood. The composition of blood varies a great deal depending on the individual, but as an average 54% is made up of a fluid called plasma while the other 46% is made up for formed elements. The percentage by volume of blood made up of formed elements has important clinical considerations; in light of this the term Hematocrit (Hct) has been adopted. Plasma is a mixture of water (92%), solutes (1%) and suspended proteins (7%). The formed elements contain cells (White and Red blood cells) and cell fragments (Platelets). Of the cells in blood, the majority are Red blood cells, making up 99.9% of all cells in the blood. The five physiological functions of blood in the human body are: regulation of pH and Ion composition, the restriction of fluid losses at injury sites, defence against toxins, and the stabilization of body temperature. (Martini, et al., 2009)

Red Blood Cells

Red blood cells (erythrocytes) are biconcave disks designed to maximize surface area for cross membrane transport. The average diameter is 7.8 µm, and the thickness varies from 0.8 µm at the center and 2.85 µm at the widest part. The shape of the cells has been carefully designed by evolution to carry out its five physiological functions. High surface area to volume ratio is important for absorbing and releasing oxygen from the lungs into the tissues. The concave disks allow the cells to form stacks known as a rouleaux that allows easy movement in very narrow vessels. The cells themselves are very ductile, allowing them to enter vessels smaller then the diameter (up to 4 µm). (Martini, et al., 2009)

Physiology of Blood

The primary physiological function of red blood cells is a complicated bio-chemical process that tends to be overlooked when approaching blood from an engineering perspective. It is therefore been decided to make a succinct introduction in order to illustrate the complexities of the biochemical nature of red blood cells.

Red blood cells are packed very tightly with protein molecules called hemoglobin (Hb). Because of the packed nature inside a red blood cell, the body has opted to remove most of the cellular organelles, with the consequence that red blood cells cannot replicate or repair – a job for an individual’s bone marrow. Hb is a complex protein consisting of four knotted strands (a quaternary protein structure); each strand houses a molecule of heme, an inorganic complex consisting of an iron atom. When oxygen enters the cells in the lungs the oxygen associates with iron, which changes the charge structure of the heme molecule. This causes a conformational shift in Hb, allowing other oxygen molecules to interact with the 3 auxiliary heme units more readily, this is referred to as cooperative ligand binding. Tissues with high oxygen demands release large quantities of carbon dioxide, which tends to lower the pH of the tissue. When an oxygenated Hb is exposed to this acidic condition (high H+ concentration), the H+ ions bind with heme causing another conformational change in the protein shape. The affect of pH on the structure of molecules is known as the Bohr affect.  This decreases the affinity of Hb to oxygen (leading to a deoxy Hb structure), which by evolutionary demand has higher affinity for carbon dioxide than oxygen. This allows for blood to deposit oxygen used for cellular consumption while absorbing carbon dioxide. (Pratt, et al., 2004)

Blood Flow

Blood is composed primarily of water and is pushed through a very extensively branched network of vessels. The vessels leaving the heart by the systemic circulation leave the heart through the ascending aorta, which quickly branches into two vessels feeding the head and the rest of the body. The systemic circulation starts off with highly elastic arteries with a mean diameter of 1.5 cm. Elasticity of the vessels is a property of rings of smooth muscle that surround the vessels. The smooth muscle assists in the propagation of the pressures wave developed from the heart. The system gradually reduces the size and elasticity of the vessels as the blood makes it way to the tissues (see table 1). As the vessels reach their respective tissue the flow is required to slow down to appropriate speeds (0.07 cm/s is recorded as a peak capillary velocity) to transfer materials through diffusion to interstitial fluids. Blood is then pushed back to the heart through veins by a small remaining pressure gradient, and in long limbs the strategic position of veins and one way valves in such a way muscle contractions can nurse the blood along. Veins, unlike arteries do not illustrate a clearly defined circular shape due to a lack of connective tissue and smooth muscle. Once the blood has returned to the heart it enters from the right side, and is pushed at a much less dramatic pressure into the pulmonary circuit. The pulmonary circuit takes the deoxygenated blood to the lungs to be oxygenated. After this journey the blood is taken back into the heart to be driven by a major pressure wave into the systemic circulation.

Also of critical importance is that the flow rate through the smaller vessels is completely controlled by a control system. Sphincters placed at the start of most capillary beds control the flow of blood, thus preserving nutrients for needy tissues.

Mathematical Description of Flow

Blood flow tends to be complex, and depending on the complexity of the model different phenomon can be illustrated. This is a consequence of the complex nature of the viscosity, dynamic pressure gradients, and cellular activities. That being said, it is usually assumed that the flow can be modelled like flow in a pipe driven with a constant pressure. The solution to the Navier-Stokes equation in cylindrical coordinates for pipe flow is typically referred to poiseuille flow [1]. See the appendix for full solution.

This provides a parabolic velocity distribution proportional to the viscosity. It is shown experimentally that poiseuille flow is a good approximation at very low hematocrit as shown by (Choi, et al., 2009). Choi used a new method of holographic microscopy to measure velocity profiles at a hematocrit of 0.05%, see figure 1.

Velocity of Red Blood cells of volume concentration of 0.05% (Choi, et al., 2009)

Blood Pressure

Blood pressure is supplied by the heart through cardiac contractions. Extensive research has been conducted modeling the pressure waves and flow rate in vessels – the most common is a method using Fourier analysis comparing harmonics to the recorded wave form. Furthermore a linear model (Windkessel Model) using a relation between pressure gradient, loss coefficient, flow rate, and the volume of the pump chamber of the heart can be used to derive a theoretical wave speed (Caro, et al., 1978).

The maximum pressure in an artery is roughly 170mmHg (22.7 KPa) vs. 80mmHg ((10.6 KPa) for veins. It is a noteworthy that the maximum pressure reached in the systemic system does not occur adjacent to the heart, but at a distance away. This is due to relatively large decrease in diameter of the vessels, and the wave propagation assisted by the smooth muscle tissues. Figure two illustrates the pressure gradient throughout the circulation.

When a patient sees a doctor he takes two blood pressures, the systolic and diastolic pressure. The blood pressure measurement technique is based on

sound, and tends to be very subjective to the machine/doctor. The procedure

Pressure distribution in blood vessels (Caro, et al., 1978)

starts with the placement of a cuff that is inflated with air. The cuff is inflated such that it cuts of the arterial pressure. The pressure is then slowly released from the cuff and the sounds coming from the vessel are monitored. The first squishing sound heard illustrates the pressure where the vessel can just open during systole (contraction of the heart), this is called the systolic pressure. The diastolic pressure is recorded when the sound stops, meaning the blood has freedom to pass through the vessel. The diastolic pressure is during the filling (diastole) phase of the cardiac cycle (Noordergraaf, 1978).

Cells in Fluid Flow

The movement of cells during flow is a complex process that is dependent on the shear rate and deformability of the suspended cells. Cells in fluid flows have commonly been modelled as spheres with no cell-cell interactions. And likewise if we follow this logic the mathematical and mechanical descriptions of particles in flow are easily discussed. Firstly the settling of cells in still fluids has important clinical impacts, as the gravity sorting of whole blood constituents has many medical uses. Stokes law [2] describes the settling velocity to be 3.74 μm/s, or about 10mm/hr. The sedimentation of red blood cells (ESR), although making the assumption cells are rigid spheres, has become an important indicator for some medical conditions. In order to conduct the test the hematocrit must be reduced one hundred fold, and the plasma must be prevented from making air contact – where proteins would initiate coagulation.

Where:

The movement of suspended spheres in pipes offer a description on the translation and rotational behaviour of dilute concentrations of red blood cells. Figure three shows the movement of such spheres and the obvious constant rotation of a suspended sphere due to shear gradient across its surface. True blood cell rotation is not constant, due to fluctuating shear stress on the non-spherical shape as the cells angular orientation changes. Fluctuations in red blood cell angular orientations appears to be a function of the shear rates, and at high values (100/s) the cells tend to remain at a position where they are parallel to the axis of the flow

Spherical particle in pipe

Another key difference between solid spheres and red blood cells is the potential for red blood cells to deform under the applied stress. Figure four illustrates the deformability of red blood cells in flow. This allows the cells to penetrate capillaries of sizes much smaller then the size of a red blood cell itself.

A key difference in modeling between red blood cells and spheres is the radial migration of the particles during flow. At low flow rates rigid spherical particles tend to remain at there relative positions, where something deformable tends to migrate to the pipe axis (Caro, et al., 1978). This is attributed to the viscous forces of the fluid dominating the movement of the particles. On the other hand at high flow, both spherical particles and cells, tend to migrate to a equilibrium position of 0.6 times the pipe radius. The equilibrium position is a function of the viscous effects of the fluid, the inertial effects of the particles, and the effect of the wall. Regardless of circumstance, when considering red blood cells there is a tendency to migrate away from the wall, which is commonly referred to as the “tubular pinch effect” (Caro, et al., 1978). Figure five illustrates the chaotic behaviour of red blood cell movement through a vessel. We see there are large fluctuations in position around the equilibrium position, but a general migration towards to this state. The red blood cell near the axis has little fluctuations in position – due to the minor shear gradient over its surface, but has a radial migration towards to the equilibrium position.

Radial position of red blood cell in vessel, (a) is the outer diameter (Caro, et al., 1978)

Several other key ideas are note worthy when considering the movements of cells through the circulation. Firstly the outer layer of a vessel tends to have lower concentrations of cells then the rest of the vessels; this arises from the geometric impossibility of maintaining concentrations right to the perimeter of a vessel. This results in a cell free layer, which at appropriately small diameters of vessels creates a lubricating layer, resulting in an overall lower viscosity than expected. A second important distinction that arises as a consequence of the cell free layer is the notion of a dynamic hematocrit. The dynamic hematocrit represents the hematocrit of the blood moving greater then the mean velocity of the vessel, where the static hematocrit would be the hematocrit of blood at rest. The end result of this statement is that overall blood cells tend to reach their final destination faster then the plasma that surrounds them. This is a consequence of the parabolic velocity profile of poiseuille flow and the cell free layer.

Up to now we have only considered flows with low concentrations of suspended particles, this is due to the complexity of the particle interactions. Although red blood cells do not have any explicit cellular interactions in the form of flagellum or “fingers” the cells tend to aggregate in a very specific manner. This manner is a specific stacking arrangement called a rouleaux, where the cells align with there larger sides touching adjacent cells (figure 6). This formation has been shown to form at shear rates less that 50/s, otherwise the formation is broken by the high shear stress developed between cells (Caro, et al., 1978).

: Rouleaux formation of red blood cells flattened out by smear, University of Wisconsin Internal Medicine website.

When considering blood we must consider large concentrations of suspended particles. This puts us in the realm of larger particle-particle stresses and an increase in: inter-particle stresses, fluid stresses as result of the no slip condition on the particles, and of cell deformation.  A measurement technique when looking at normal hematocrit levels is the use of ghost suspensions, where 95% of the cells in solution can be died transparent. This allows us to measure velocity profiles of cells at physiological hematocrit levels. Typical measured velocity distributions are shown in figure seven. The increased concentrations of hematocrit flatten out the velocity distribution curve, creating a

larger portion of the flow sustaining a constant velocity. This is illustrated in figure seven where we see at large concentrations spherical particles become completely plug like. Cells (line 2) have a flattened curve, where line three (liquid drops) follow poiseuille flow exactly. This leads to an important distinction between our suspensions of spheres vs. red blood cells. An interesting note: suspensions of spheres become nearly solid at a concentration of 60%, where blood still flows up to 98%.

Velocity profiles of ghost solutions at physiological concentrations (Caro, et al., 1978)

The flattening out of the velocity distribution seems to only happen at relatively low shear rates. At higher shear rates the particle-particle interactions seem to be overshadowed by the viscous effects of the fluid. Therefore at high shear rates the viscosity of the fluid does not depend on the shear rate, thus allowing blood to be quasi-Newtonian.

Viscosity

As we have seen the viscosity of blood is a function of the hematocrit, and shear rate of the flow. This immediately indicates that the fluid, over the entire range of flow conditions, is non-Newtonian. This is illustrated in figure 8, showing the apparent viscosity of the fluid. As previously discussed we can see that the viscosity becomes independent of the shear rates when the rate of shear is greater than 10/s The figure represents the nature of whole blood, which itself is made up of a suspension of cells and a Newtonian fluid plasma (Caro, et al., 1978).

The apparent viscosity of blood at varying shear rates (Caro, et al., 1978)

The first theoretical formula to discuss the viscosity of a suspension of particles was published in 1906 by Albert Einstein. He used mechanics to deduce an expression for viscosity as a linear function of the concentration for suspended particles, see equation 3. Unfortunately his formulation has little relevance in our discussion, as his formulation assumed spherical particles in very dilute concentrations.

where μs = suspension viscosity, μf = fluid viscosity, c is volume fraction of particles.

In the last half a century the field of suspensions of particles has exploded, particularly with the groundbreaking paper published by R.A Bagnold in 1954 titled “Experiments on a Gravity Free Dispersion of Large Solid Spheres in a Newtonian Fluid under Shear”. Bagnold discusses three modes of sediment transport: The Grain-Inertia Region, the Macro-Viscous Region, and the transition region. He quantifies flows into their appropriate categories by a ratio of the inertial stress to the viscous stresses in the fluid flow (equation 4) – this would later come to be known as Bagnold’s number. The grain region is defined as Ba>450, and the viscous region is defined as Ba<40 (Bagnold, 1954). His argument should not be surprising as it follows the same logic to characterize the turbulence using the Reynolds number.

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